This chapter introduces the Discrete Fourier Transform (DFT) and
points out the mathematical elements that will be explicated in this
book. To find motivation for a detailed study of the DFT, the reader
might first peruse Chapter 8 to get a feeling for some of the many
practical applications of the DFT. (See also the preface on page
.)

Before we get started on the DFT, let's look for a moment at the
Fourier transform (FT) and explain why we are not talking about
it instead. The Fourier transform of a continuous-time signal
may be defined as

Thus, right off the bat, we need calculus. The DFT, on the
other hand, replaces the infinite integral with a finite sum:

where the various quantities in this formula are defined on the next
page. Calculus is not needed to define the DFT (or its inverse, as we
will see), and with finite summation limits, we cannot encounter
difficulties with infinities (provided
is finite, which is
always true in practice). Moreover, in the field of digital signal
processing, signals and spectra are processed only in sampled
form, so that the DFT is what we really need anyway (implemented using
an FFT when possible). In summary, the DFT is simpler
mathematically, and more relevant computationally than the
Fourier transform. At the same time, the basic concepts are the same.
Therefore, we begin with the DFT, and address FT-specific results in
the appendices.